Bounds in cohomology
type: Item
instance of: scholarly article
title: Bounds in cohomology
zbMATH Open document ID: 0953.13005
DOI: 10.1007/BF02773216
author: Hans Schoutens
published in: Israel Journal of Mathematics
publication date: 2000-09-19T00:00:00Z
review text: The author introduces a measure of complexity for affine algebras and their finitely generated modules. For some affine local \(K\)-algebra \(R\) and a finitely generated \(R\)-module \(M\) with the presentation \(R^s @>f>> R^t \to M \to 0\) the complexity of \(M\) is at most \(d\) provided \(t, s \leq d\) and each entry of the matrix of \(f\) is the quotient of two polynomials of degree at most \(d.\) Then the author studies various cohomological operations and numerical invariants, uniformly bounded with respect to these complexities. The Lefschetz principle, most succinctly stated, is the following isomorphism of fields \[ {\mathbb{C}} \simeq \prod_U {\mathbb{F}}_p^{\text{alg}}, \] where \({\mathbb{F}}_p^{alg}\) denotes the algebraic closure of the \(p\)-element field \({\mathbb{F}}_p\), \(p\) a prime number, and \(U\) the non-principal ultrafilter on the set of primes. This enables the transfer of properties between various characteristics. The first application concerns the Bass conjecture and the ``new intersection theorem''. Its validity for affine local rings over an algebraically closed field of characteristic zero can be derived from its validity in the case of positive characteristic. -- The second application concerns the Zariski-Lipman conjecture in positive characteristic. While the conjecture is not true in positive characteristic, as shown by \textit{J.~Lipman} [see Am. J. Math. 87, 874-898 (1965; Zbl 0146.17301)], the author proves the following result in positive characteristic: Let \(R\) denote the local ring of a point \(P\) on a hypersurface over an algebraically closed field \(K\) such that the module of \(K\)-invariant derivations on \(R\) is free. Then \(P\) is a non-singular point, provided the characteristic is larger than some bound depending on the degree of the hypersurface. Reviewers remark: The new intersection theorem as well as the Bass conjecture is true in general as was finally shown by \textit{P. Roberts} [see C. R. Acad. Sci., P…
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D22
zbMATH DE Number: 1445778
zbMATH Keywords: complexity of affine algebras
zbMATH Keywords: local algebra
zbMATH Keywords: constructibility
zbMATH Keywords: Zariski-Lipman conjecture
zbMATH Keywords: Lefschetz principle
zbMATH Keywords: Bass conjecture
zbMATH Keywords: new intersection theorem
zbMATH Keywords: characteristic zero
zbMATH Keywords: positive characteristic
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1445778